Key goals of this lecture (and accompanying self-study)…

0. Get excited

1. What’s this module’s purpose and story arc? How to get the most out of it.

2. What’s Microeconomics? Why is it useful? Why use ‘models’? Some applications of models

• Recap microeconomics examples, get econ brain flowing

Enrichment texts

Have a look at in case you are thinking of going for a PhD in economics or taking further coursework at the MRes level

1. Microeconomic Theory: Basic Principles and Extension (also by Nicholson and Snyder): more maths and extensions

2. David Autor’s MIT Open Courseware “Microeconomic Theory and Public Policy”: More rigorous, close to PhD level, includes good research applications; we’ll be drawing from it

3. PhD level: Jehle and Reny’s “Advanced Microeconomic Theory”, or Hal Varian’s text “Microeconomic Analysis”; Mas-Colell et al for wizards only

1. Economic basics (weeks 1-2)

• Economic models, maths tools, introduction (NS ch. 1)

• Utility, preferences, indifference curves, budget constraints (NS 2)

we build it up and then burn it down…

2. Build the model, put it together, examine it (weeks 3-5)

The Demand side:

Demand curves: Individual and market demand (NS ch. 3)

• Production, costs, returns to scale, choice of inputs

Profit maximisation and supply, perfect competition in a single market

Supply curves, entry/exit, Consumer and Producer Surplus, general equilibrium and welfare (brief)

3. How the market can go wrong (and how to maybe fix it) (weeks 6-7)

• Market failures – Public goods

• Monopolies; price discrimination as an imperfect remedy

4. Extensions to the model and applications (weeks 8-11)

• Uncertainty (basic concepts, EU, risk aversion, investment choices)

• Game theory; experimental evidence

• `Behavioural’: Limits to cognition, willpower, self-interest; applications and evidence

The big (small) question

What’s gonna be on the exam??

• Everything

• I reward broad understanding; ‘can YOU explain it?’
  • Show strengths in at least some areas (maths, readings, explanation…)

Examples of exam material: Practice problems in notes/lectures, problem sets, mock and sample questions on the VLE; mock exams, previous exams (esp since 2015-16)

##But anything I say you can find online, or in a book, so why are you here?

To interact

Not to hear what the *lecturer has to say, but for the lecturer to respond to what

Resources

The VLE and other resources

• VLE let’s see it

• Ask questions and make comments on the forum

• LINK, I will monitor it

• I may not have time to give a detailed answer to all individual emails; but you are welcome to come to my office hours –>

The ‘HANDOUT’ file/web-book

For now, available to you exclusively on the VLE

Contains all the lecture slide material and more, html links, etc.

• As we progress…
  • Read the introduction: Key information to understand how it works (abbreviations etc)
  • Leave comments/question directly on file via hypothes.is; it will be a conversation!

In-lecture interaction

• In-lecture experiments and games

• Ask me questions throughout the lecture

• raise a white handkerchief if you are lost

• Draw-along and solve-along

• I will hot and cold-call

How to do well

1. Put in the work

• Read in advance, bring questions (can we flip?)

2. Care about the content

3. Interact and discuss: Peers, Forum, office hours

4. Interpret the exam instructions correctly! (and use your time wisely)

–>

Fortune Cookie Wisdom

Heavies

Economists do not know everything (but we have thought through many arguments)}

Most non-economists do not fully understand these arguments, and they make mistakes, and they worry.

• But sometimes ignorance is bliss.

… more deep truths to follow, enough for now

\(\rightarrow\) ‘All you can eat’ \(\rightarrow\) Spotify, Netflix, Kindle Unlimited

Imperfection: Lack of information about ‘experience goods’, lack of trust in one-shot-interactions

\(\rightarrow\) ??

\(\rightarrow\) ??

\(\rightarrow\) Uber, AirBnb, ‘bilateral reputation systems’

Shyness and fear of ‘losing face’

• \(\rightarrow\) Tinder, Squad, etc.

Ties in to my research …

What is Economics?

‘Economics is the study of the allocation of scarce resources among alternative uses.’

‘Economics is the study of mankind in the ordinary business of life.’ Alfred Marshall

What is Microeconomics?

The study of the (economic) choices individuals and firms make and how these choices create markets.

Largely, using theoretical and mathematical ‘models’ that depend on strong assumptions.

What do models give us?

There are different views of this

Assumptions \(\rightarrow\) Results

and sometimes \(\rightarrow\) testable predictions (if the assumptions hold)

So why learn these models?

• A starting point

• (Sometimes) make testable concrete predictions

• Building insight, clear arguments, a way of thinking

• Discussion is framed around them; seen as a ‘baseline’

• Understand the models to effectively critique or extend them

Differing views on the use of economic models

Instrumentalist:

The Methodology of Positive Economics (Friedman): the ultimate goal of theory is to “yield valid and meaningful … predictions about phenomena not yet observed”

‘Fictionalist’ (Sugden):

describes a fictional world that is credible or truthlike in something like the way that the events of a realistic novel are; the model connects with the real world by relations of similarity

Are these models predictive?

If not, are they useful?

Economic Models

The PPF: a ‘model’ and a way of seeing things

Principle 1: Scarce Resources

Principle 2: Scarcity involves opportunity cost.

Above PPF: opportunity cost of more clothing is less food.

Principle 3: Opportunity costs are (often) increasing

• ‘Law’ of diminishing marginal returns

Akerlof’s Lemons model (simplified)

See notes here

Application 1.1: Economics in the Natural world

Studies of honeybees have found that they generally do not gather all of the nectar in a particular flower before moving on.

Why not?

• Consider the same for the UK/Exeter; give your best estimate

• How does the analysis differ from the one your uncle would do?

Hey, ma and pa, what determines the price of a bread and the amount that gets sold?

Would an increase in the minimum wage ‘help the poor’?

See Autor lecture 1: the minimum wage debate

Adam Smith and the Invisible Hand

Prices

• Not random nor morally determined

• Signals to direct resources, reflecting the ‘worth’ of goods

Labour-cost-based theory of prices

Claim: If it takes twice as long for a hunter to catch a deer as to catch a beaver, one deer should trade for two beavers.

Why/when should this not hold (or not tell the whole story)?

Farmland was expanding in England … as new and less fertile land was brought into use,

it would naturally take more labor … to produce an extra bushel of grain.

Ricardo - Diminishing returns: the cost of producing one more of good A (here, food)—in other goods foregone—rises as more of A is produced

Diminishing returns/increasing costs:

• Explains why a supply curve might slope upwards
  • Produce more \(\rightarrow\) costs more \(\rightarrow\) higher price necessary

• Left unexplained: How much is produced, thus ‘what is the price?’

Upwards-sloping supply curve, but where on this curve do we end up?

Draw: the famous Marshallian cross

Can you explain?

• The inefficiency of any price other than where \(Q^D(p)=Q^S(p)\)?

• If the price was set at a different value, what forces might push it to the equilibrium?

• Who gains and who suffers with a government-imposed price floor/ceiling?

APPLICATION 1.4: Economics According to Bono

Do US farm subsidies help or hurt Africans in net?

Consider the effects on African farmers and African producers. How could we consider the `net effect’?

Review questions and problems from Chapter 1

Some questions/problems I liked are in the handout… also see Problem Set 1 on the VLE

Be sure that you can do problems like 1.1 in the N&S text without difficulty;

• For 1.1.C, note that for the supply curve, quantity supplied is never negative – below a certain price, it will just be zero.

• Also consider ‘review questions’ 6 and 8 from the text

• will go over the key parts of these problems, and you can ask questions

Goals of this lecture (and accompanying self-study)

• Overview of (re)-aquaintance with maths tools we will use

• Flavour of what empirical microeconomics is, key issues in empirical work

Covers:

• Nicholson/Snyder Chapter 1a: Mathematics used in Microeconomics

A very good resource – Khan academy: https://www.khanacademy.org/math/

• E.g., see their Introduction to differential calculus

##Simple stuff

(Univariate) Function

A ‘map’ from one or more variables \(x\) to an outcome \(y=f(x)\)

• for each value of \(x\) the function tells you a single value of \(f(x)\); typically we assign \(y=f(x)\)

Linear function

A function of the form \(y=a+bX\); e.g., \(y = f(x) = -10 + 3x\)

• Plotted as a straight line; intercept \(a\), constant slope \(b\)

Slope of \(y = f(x)\)

The change in y for a given change in x. ‘Rise over run’ \((\Delta y / \Delta x)\).

• Arc slope, point slopes

Nonlinear (univariate) function : A function \(f(x)\) of a form other than \(f(x) = y=a+bX\);

• E.g., a quadratic function \(y = f(x) = a + bx + cx^2\)
  • E,g, \(y = f(x) = 10 - 2x + 3x^2\)
• Or a logarithmic \(y=ln(x)\) or exponential function \(y = exp(x)\)

For linear functions the slope is the same at any point. For nonlinear functions it may differ at each point.

Instantaneous rate of change (instantaneous slope)

The slope of the line tangent to the curve at a single point

• Convex function: Slope everywhere increasing, unique minimum where slope \(=0\)
• Concave function: Slope everywhere decreasing, unique maximum where slope \(=0\)

Derivative of a function

A derivative of a function \(f(x)\) is another function called \(f'(x)\). \(f'(x)\) tells us the (point) slope of the function \(f(x)\) at any point \(x\).

• For example, the derivative of the function \(f(x) = 2x + 3\) is \(f'(x) = 2\)
  • For this linear function the slope is a constant, 2

• E.g., the derivative of the quadratic function \(f(x) = x^2 -4x + 1\) is \(f'(x) = 2x - 4\)

• derivative of the quadratic function \(f(x) = x^2 -4x + 1\) is \(f'(x) = 2x - 4\)

• E.g., the slope at \(x=1\) is $f’(x;x=1) = 1*1 - 4 = -2

• The slope is zero where \(f'(x)=2x-4=0\), or where \(x=2\)

  • Is \(x=2\) at a min, a max, or neither? How do we know?

Minimum, maximum, or neither?

• \(f'(x)\) is a function that tells us the slope of \(f(x)\), or how \(f(x)\) changes in \(x\) at any point \(x\)
• In turn, the derivative of \(f'(x)\) is called \(f''(x)\).
  • Tells us how the slope changes as \(x\) increases

Oversimplifying:

• slope always increasing \(\rightarrow\) \(f''(x)>0\) everywhere \(\rightarrow\) convex (u-shaped) function \(\rightarrow\) single where \(f'(x)=0\)

• slope always decreasing \(\rightarrow\) \(f''(x)<0\) everywhere \(\rightarrow\) concave (inverse-u) fncn \(\rightarrow\) single where \(f'(x)=0\)

Functions of two or more variables (multivariate functions)

Utility, profit, cost, production, returns, etc.

• May depend on multiple variables/inputs
• Need to illustrate tradeoffs between these

\[y=f(x,z)\]

• \(y\) may increase and/or decrease in \(x\) and in \(z\),

• The rate of increase of y in \(x\) may depend on the values of \(x\) and \(z\)

  • Similar for the rate of increase of y in z

E.g., \[y=\sqrt(xz) = x^{1/2}z^{1/2}, x \geq 0, z \geq 0\]

Projecting a function up from X,Y space into the Z axis:

Contour lines (we will come back to this later!)

Contour lines

Level sets - depict combinations of variables that hold the function constant at a particular value

f(x,z) = A for some value \(A\)

Level sets: E.g., indifference curves, isoquants and isocost curves.

Contour lines on a map

Consider a production function:

\[Y = f(K,L) = \sqrt(KL)\]

Setting this equal to 1 we can map out ‘all combinations of K and L that produce output \(Y=1\)’:

\[ Y = \sqrt(KL) = 1 \rightarrow KL = 1 \]

\[ \rightarrow K = 1/L \]

Setting this at Y = 2

\[ Y = \sqrt(KL) = 2 \rightarrow KL = 4 \] \[ \rightarrow K = 4/L \]

Simultaneous equations

E.g.,

\[ X + Y = 3 \] \[ X - Y = 1 \]

Holds only where \(X=2, Y=1\), the ‘unique solution’

Meaningless to ask ‘how does a change in X affect Y?’ in the above context.

Empirical microeconomics/econometrics

Empirical research

Uses evidence from the real world, i.e., observation, to answer questions

Econometrics

The ‘science’ of using data to answer economic questions. Uses statistical tools and often economic theory

Micro-data

Data where the unit of observation is an individual, household, firm, etc.

See, e.g.,

• Euro-stat microdata

• Data on individual donations (at the donor level) from a large scale field experiment (Karlan and List) downloadable HERE

• Price and product data on gluten-free bread at the supermarket level

Empirical(ish) example

Trying to estimate demand curve, hypothesize linear function \[Q_d = a-bp\]

Suppose we know price is shifting because of costs, shifts in supply curve, or the firm experimenting

Observe price & quantity data for a period where ceteris paribus is reasonable

Fit ‘best’ line (minimise error) through these points

• Estimate demand slope & intercept, use to make inferences

• Never fits exactly. why not?

• Advanced: Why is this *only meaningful if we are observing shifts in the

Ceteris paribus

All [most] economic theories employ the assumption that ‘other things are held constant.’

• Above, demand may differ between weeks/stores, weather changes, etc.

the points may lie on several different demand curves, and attempting to force them into a single curve would be a mistake.

\(\rightarrow\) Carefully ‘control’ for other observable factors (a partial solution)

Exogenous and endogenous variables

Suppose a model of supply and demand with ‘exogenous shifters’:

• \(W\): an external or exogenous variable that increases the quantity demanded (at any price)

• \(Z\): an exogenous variable that makes production more costly

Now write the supply/demand model as: \[Q=-P+W\] \[P=Q+Z\]

• P and Q are endogenous variables, so don’t ask ‘how does P change in Q?’
• W and Z are exogenous variables
  • If we know W and Z we can solve for the equilibrium Q and P
  • ‘Comparative statics’: how equilibrium responds to shifts in W & Z

Application 1A.3: … Changing world oil prices (time-permitting)

Empirical work has estimated:

\[ Q = 85 - 0.4P \: (D) \] \[ Q = 55 + 0.6P \: (S) \]

Solving: \(85 - 0.4P=55+0.6P \rightarrow P = 30, Q = 73\)

Approximates 2000-2002 price

Why did price rise to US$130 in 2008 and fall to $50 by March ’09?

• China & India’s economies grew \(\rightarrow\) growth in the world economy by 3-4% per year
  • (Various calculations) \(\rightarrow\) Demand shifts out from \(Q_D = 85 - 0.4P\) to:

\[Q_D = 112 - 0.4P\] \[Q_S = 55 + 0.6P\]

\(\rightarrow\) solves to \(P=57, Q=87\)

• Overall price inflation, US$ devaluation \(\rightarrow\) gets us to about $94. So why the 130 USD price?

##First problem set: coverage

See the Problem set 1 file on the VLE

• Representative answers for each problem set given about 1-week after posting

• Five support classes (tutorials), cover parts of these

E.g., …

1. Plotting supply and demand “for orange juice”, solving, for equilibrium price, excess demand/supply at non-equilibrium prices

2. Impact of shift in demand (for milk) on equilibrium, depending on slope of supply curve, explain

3. Example of some (tricky) MCQs from previous exams

E.g., “True or false: It is valid to plot observed prices and quantities traded in a market and fit a line through them to estimate a market demand curve.”

4. Discussion questions: practice writing concise essays and bullet points

Motivation (preamble)

Consider a decision you recently made?

• Define this decision clearly.

• How do you think you decided among these options?

2 minutes: discuss with your neighbour

Suppose I asked you

‘State a rule that governs (determines/characterizes) how people do make decisions’…

I want this rule to be…

1. Informative (it rules out at least some sets of choices)

2. Predictive (people rarely if ever violate this rule)

Similar question:

‘State a rule that governs how people should make decisions’..

By ‘should’ I mean that they will not regret having made decisions in this way.

2 minutes: discuss with your neighbour

If people did follow these rules, what would this imply and predict?

Rules defined as ‘axioms about preferences’

‘Standard axioms’ \(\rightarrow\) (imply that) choices can be expressed by ‘individuals maximising utility functions subject to their budget constraints’

\(\rightarrow\) yields predictions for individual behavior, markets, etc.

• ‘Budget constraints’, compute and model them

• Maximising utility subject to constraints
• optimisation condition for this

<!---
price ratios \& mgnl rates of substn.
-->

• Depict and interpret optimisation with indifference curves and budget constraints

Main readings and sources

• McDL: Ch 12, plus ‘preference relations’

• AUT: Lecture 3 - Axioms of Consumer Preference and the Theory of Choice (note, slightly different notation)

• NS: Ch 2

Enrichment and extensions (optional)

David Autor’s notes – MIT Microeconomic Theory and Public Policy Open CourseWare

How is this used?

• Utility will be expressed as a single number that arises from the combination of all goods and services consumed.

Essentially, economists assume that `when making a choice among all available and feasible options, an individual will choose the one that yields the greatest utility’

Utility from two goods

\[Utility = U(X,Y; other)\]

• Leisure and ‘goods consumption’
• Food and non-food
• Coffee and tea (holding all else constant)

Measuring and comparing utilities

Utility is not ‘observable and measurable in utils’

(Unlike midi-chlorians or thetans)

• Utility is seen to govern an individual’s choices and thus it’s only inferred indirectly, from the choices people make

Revealed preference: if Al buys a cat instead of a dog, and a dog was cheaper, we assume Al gets more utility from a cat

Interpersonal comparisons are difficult

• Who gets ‘more’ utility?

…Interpersonal comparisons are difficult

• Transfer from Al to Betty: Is the reduction in Al’s utility more or less than the increase in Betty’s?

Standard utility functions are only ‘identified’ (only make distinct predictions) ‘up to a monotonic transformation’

• i.e., a ‘rank preserving’ transformation.

• E.g., multiply by a positive number, log, square, add/subtract from it;

  • the new utility function makes the same predictions; this is not helpful, because it means we cannot distinguish these empirically!

• The “benefit”: we can transform (e.g., log) a utility function for convenient calculation

• When we deal with uncertain choices, the ‘Von Neumann - Morgenstern’ utility functions will be somewhat more restrictive and thus better identified.

1. Completeness

\[A \succ B, B \succ A, \: or \: A \sim B \]

Fancy notation: Either A preferred to B, B preferred to A, or A indifferent to B

Forbidden: “I can’t choose between a ski holiday and a beach holiday, but I am not indifferent”

2. Transitivity

\[ A \succ B \: and \: B \succ C \rightarrow A \succ C \]

• Similar for indifference (\(\sim\))

If I prefer an Apple to a Banana and a Banana to Cherry

then I prefer an Apple to a Cherry.

If not \(\rightarrow\) money pump.

3. More is better (similar to nonsatiation, ‘monotonicity’)

Draw this!

ootnotesize{If the product is a ‘bad’ (e.g., pollution), redefine as the absence of the product}

Continuity and nonsatiation

• How can we compare the “?” areas? Which are preferred?

\(\rightarrow\) Compare utilities, depict using Indifference Curves

Credit: www2.econ.iastate.edu

Credit: Frank’s Economics on the web (MIT)

Q1: How do we know \(E \succ B\) ?

Q: How do we know \(E \succ A\) ?

Why ‘voluntary trade’?

The indifference curve offers some intuition.

Back to fig 2.2 (board or visualiser)

• A to B: willing to give up 2 hamburgers to get 1 more soda.

\(\rightarrow\) slope \(-2\), \(MRS=2\)

• From B to C? (think about it)

…willing to give up 1 hamburger to get 1 more soda \(\rightarrow MRS =1\)

• C to D?

\(MRS = \frac{1}{2}\)

• Note the decline: ‘diminishing MRS’: may reflect satiation

More formal concept: (strictly) convex preferences \(\Leftrightarrow\) (strictly) ‘quasiconcave’ utility function, allowing ‘n goods’:

With diminishing MRS, for a ‘convex combination’ of the two bundles \(\mathbf{A}=(X_A,Y_A)\) and \(\mathbf{D}=(X_D,Y_D)\)

\[U(\alpha\mathbf{A}+(1-\alpha)\mathbf{D})\geq\alpha U(\mathbf{A}) + (1-\alpha)U(\mathbf{D})\]

where \(0<\alpha<1\).

(Note switch to a ‘weak inequality’ here for quasiconcavity, strict for ‘strict quasiconcavity’)

e.g., (with two goods),

\[U(\alpha X_A + (1-\alpha)X_D,\alpha Y_A + (1-\alpha)Y_D)\geq \] \[\alpha(U((X_A,Y_A)) + (1-\alpha)(U(X_D,Y_D))\]

App 2.3: Product positioning in marketing (read at home, see handout)

• Preference for balance? (Convex indifference curves)?
  • If market research suggests a broad group indifferent between A and D (in fig 2.3), they may strictly prefer G
  • \(\rightarrow\) a possible niche for a new profitable product

• Utility/indifference curves: Also a framework for marketing analysis

Perfect complements

Goods A and B are Perfect Complements when an individual only gains utility from (more) A if she also consumes a defined (additional) amount of B, and vice-versa

These goods are ‘enjoyed only in fixed proportions’.

Choices are subject to constraints :(

You cannot spend more than your (lifetime) income/wealth \(\rightarrow\) budget constraint.

Budget constraint algebra

If I spend all my income (I will do over a ‘relevant lifetime’):

Expenditure on X + Expenditure on Y = Income (I)

\[P_X X + P_Y Y = I \]

To see how \(Y\) trades off against \(X\), rearrange this to:

\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]

• Intercept \(\frac{I}{P_Y}\): amount of Y you can buy if you only buy Y
• Slope \(-\frac{P_X}{P_Y}\): how much Y you must give up to get another X

Utility maximization

If slope budget constraint = slope indifc curve at point X,Y \(\rightarrow\) \[P_X/P_Y = MRS(X,Y)\]

Warning: This equality holds at an optimal choice; it doesn’t hold everywhere.

At an optimal consumption choice (given above assumptions)

• Consume all of income; locate on budget line; follows from ‘more is better’

• Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods

Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods

Key intuition:

If I can give up X for (buy less X, get more Y) at some rate, and the benefit I get from doing this is at a different rate,

…then I can make myself better off.

Thus the original point could not have been optimal.

More insight

Recall \(U=U(X,Y)\).

\[U_X(X,Y) := MU_X(X,Y)\]

Derivative w/ respect to X: rate utility increases if we add a little X, holding Y constant

• Similarly for \(MU_Y\).

MRS: ‘how much Y would I be willing to give up to get a unit of X’?

Ans: Depends on marginal benefit of each … we can show \(MRS(X,Y)=\frac{MU_{X}}{MU_{Y}}\)

The ‘first order change in utility’ (or ‘total differential’) is:

\[ dU = \frac{\partial U}{\partial X}dX + \frac{\partial U}{\partial Y}dY\] \[ = MU_{X}dX + MU_{Y}dY\]

• Where = \(\frac{\partial U}{\partial X}\) refers to the partial derivative of U(X,Y) with respect to X, and similarly for Y. I.e., the marginal utility.

Essentially, approximates the total change in utility for very small changes in X and Y. Setting it equal to zero and rearranging it yields the rate, at the margin, one is willing to give up Y for X:

\[ dU = MU_{X}dX + MU_{Y}dY = 0 \]

\[\frac{dY}{dX}=-\frac{MU_{X}}{MU_{Y}}\]

Rearranging the utility maximising condition yields more intuition:

\[P_X/P_Y = MRS = MU_X/MU_Y\]

(at each consumption point X,Y)

\[\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\]

Same ‘bang for each buck’ (if optimising)

##Caveat on ‘corner solutions’

• If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold

Advanced: This is a necessary but not sufficient condition, sufficient if DMRS everywhere

But…

But…

• If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold

• But you might consume none of some good (say X):
  • if even with no X, \(MU_X/P_X<MU_Y/P_Y\), the marginal utility of the first unit ‘per pound’ is lower

Enrichment:

Constrained problem equivalent to optimising a single unconstrained ‘Lagrangian’ function

• See Autor Lecture 4, ‘mathematical solution to the Consumer’s problem’

• This yields the above optimisation (first-order) conditions

(subject to non-negativity constraints)

##App 2.4: ticket scalping

##App 2.5: What’s a rich uncle’s promise worth?

##Using the model of choice

1. Why do people spend their money on different things?
2. What do different preferences/indifference curves imply for choices?

Move to ppt slides here beginning with ‘Utility Maximization: A Graphical View’

Perfect substitutes, but not identical, e.g.,

\[U(X,Y)=4X+3Y\]

Rates each increase utility per-unit (derivative) are constant: \(MU_X = 4\), \(MU_Y = 3\)

So (for perfect substitutes) buy the one that increases it more *per-

With perfect substitutes: ‘Bang for the buck’ rule

\[U(X,Y)=4X+3Y\]

• Compare \(MU_X/P_X\) to \(MU_Y/P_Y\)

• Here, if \(4/P_X > 3/P_Y\), then buy X

• if \(4/P_X < 3/P_Y\), then buy Y; (if equal, buy either)

• Rearranging, if \(P_X < 4/3 P_Y\), buy X … etc.

Warning: If not perfect substitutes, MU ratios depend on consumption levels.

Perfect complements

Mathematical function example:

\[U(X,Y)=min(2X,Y)\]

E.g., X: bicycle frames, Y: wheels.

Warning: this min function looks backwards, but it’s correct; see notes

• Shortcut: figure out the proportions it will be consumed in
  • determine cost of ‘1 bundle of the combo’ at given prices
  • … then buy as many such bundles as you can afford

Middle-ground (*)

A Cobb-Douglas example

\[ U(X,Y)=\sqrt(XY) \]

\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (Y/X)^{1/2}\]

\[MU_Y = \frac{1}{2} (X/Y)^{1/2}\]

Here, amount of Y you’d give up to get a unit of X:

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

{Check reasonable: The more Y I’ve , the more Y I’d give up to get another X :)}

\[ U(X,Y)=\sqrt(XY) \]

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

Calculus:

\(MU_X\) is the slope of \(U(X,Y)\) in X at a particular point, i.e., the (partial) derivative with respect to X

\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (XY)^{-1/2}Y = \frac{1}{2} (Y/X)^{1/2}\]

Similarly,

\[MU_Y = \frac{1}{2} (X/Y)^{1/2}.\]

…Cobb-Douglas ctd

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

Here utility-maximization requires, at optimal choices of X and Y: \[MRS(X,Y)= Y/X = P_X/P_Y\]

For any price ratio, find ratio of Y & X.

With prices and income, \(I\), find consumption of X & Y.

Rearranging optimization condition:

\[Y P_Y = X P_X\]

Optimization condition for this particular utility function:

\[Y P_Y = X P_X\]

Combining this with the budget constraint \[P_X X + P_Y Y = I\]

solve for X & Y, as fncns of prices & income (see notes) \(\rightarrow\)

\[Y = I/(2P_Y)\] \[X = I/(2P_X)\]

###Cobb-Douglas general form (from other text)

\[U(X,Y)=X^aY^b\]

where a, b are positive constants

This is “homothetic”:

\[MU_X=\frac{\partial U}{\partial X} = aX^{a-1}Y^b\] \[MU_Y=\frac{\partial U}{\partial Y} = bX^{a}Y^{b-1}\]

Taking the ratio of these yields

\[MRS = \frac{a}{b}\frac{Y}{X}\]

So only the ratio Y/X affects the MRS; e.g., double both and MRS is the same.

• Another relevant example: ‘Constant Elasticity of Substitution preferences’

###A non-homothetic example

\[U(X,Y)=X+ln(Y)\]

Here Y has a diminishing MU, but for X MU is constant

\[\frac{MU_X}{MU_Y}= \frac{1}{1/Y}=Y\]

So the MRS increases as Y increases, but it’s independent of the amount of X consumed.

So, if we double both then (class question)?

Ans: MRS doubles

This represents Quasilinear preferences:

More generally (and usefully) \(U=x_1+f(x_2,...,x_n)\) linear in one ‘numeraire’ good \(x_1\).

##App 2.6: Loyalty programmes (Skip in lecture)

(Doctoral-prep concepts)

• A formal definition and proof of the idea that a utility function ‘represents’ preferences

• Exercises like ‘show that lexicographic preferences cannot be represented by a continuous utility function’

What conditions on a (multivariate) utility function ensure it exhibits the equivalent of a diminishing marginal rates of substitution?

• Something called ‘quasi-concavity’, equivalent to having ‘convex upper contour sets’

Representation of the MRS between all goods as a matrix, properties of this

• Strict quasi-concavity takes the place of DMRS in ensuring ‘simple optimisation conditions’

Second problem set: covers chapter 2 (and chapter 3) – see VLE

Preferences, Utility, Consumer optimization,

individual and market demand curves.

This is a very large problem set (others are smaller).